x^2+x=90.25

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Solution for x^2+x=90.25 equation: 


Simplifying
x2 + x = 90.25

Reorder the terms:
x + x2 = 90.25

Solving
x + x2 = 90.25

Solving for variable 'x'.

Reorder the terms:
-90.25 + x + x2 = 90.25 + -90.25

Combine like terms: 90.25 + -90.25 = 0.00
-90.25 + x + x2 = 0.00

Begin completing the square.

Move the constant term to the right:

Add '90.25' to each side of the equation.
-90.25 + x + 90.25 + x2 = 0.00 + 90.25

Reorder the terms:
-90.25 + 90.25 + x + x2 = 0.00 + 90.25

Combine like terms: -90.25 + 90.25 = 0.00
0.00 + x + x2 = 0.00 + 90.25
x + x2 = 0.00 + 90.25

Combine like terms: 0.00 + 90.25 = 90.25
x + x2 = 90.25

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
x + 0.25 + x2 = 90.25 + 0.25

Reorder the terms:
0.25 + x + x2 = 90.25 + 0.25

Combine like terms: 90.25 + 0.25 = 90.5
0.25 + x + x2 = 90.5

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 90.5

Calculate the square root of the right side: 9.513148795

Break this problem into two subproblems by setting 
(x + 0.5) equal to 9.513148795 and -9.513148795.


Subproblem 1
x + 0.5 = 9.513148795

Simplifying
x + 0.5 = 9.513148795

Reorder the terms:
0.5 + x = 9.513148795

Solving
0.5 + x = 9.513148795

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 9.513148795 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 9.513148795 + -0.5
x = 9.513148795 + -0.5

Combine like terms: 9.513148795 + -0.5 = 9.013148795
x = 9.013148795

Simplifying
x = 9.013148795


Subproblem 2
x + 0.5 = -9.513148795

Simplifying
x + 0.5 = -9.513148795

Reorder the terms:
0.5 + x = -9.513148795

Solving
0.5 + x = -9.513148795

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -9.513148795 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -9.513148795 + -0.5
x = -9.513148795 + -0.5

Combine like terms: -9.513148795 + -0.5 = -10.013148795
x = -10.013148795

Simplifying
x = -10.013148795


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {9.013148795, -10.013148795}

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